MinT - Best Known (178−107, 178, s)-Nets in Base 4

Best Known (178−107, 178, s)-Nets in Base 4

(178−107, 178, 66)-Net over F₄ — Constructive and digital

Digital (71, 178, 66)-net over F₄, using

t-expansion [i] based on digital (49, 178, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(178−107, 178, 105)-Net over F₄ — Digital

Digital (71, 178, 105)-net over F₄, using

t-expansion [i] based on digital (70, 178, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(178−107, 178, 660)-Net in Base 4 — Upper bound on s

There is no (71, 178, 661)-net in base 4, because

1 times m-reduction [i] would yield (71, 177, 661)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 36883 829541 472230 926468 353103 411179 395547 610991 005196 210089 281440 682186 915778 862084 914327 859409 806864 395920 > 4¹⁷⁷ [i]